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Mathematics Question on Differential equations

For any real numbers α\alpha and β\beta, let yα,β(x),x∈Ry _{\alpha, \beta}( x ), x \in R, be the solution of the differential equation
dydx+αy=xeβx,y(1)=1\frac{d y}{d x}+\alpha y=x e^{\beta x}, y(1)=1
Let S=\left\\{y_{\alpha, \beta}(x): \alpha, \beta \in R\right\\}. Then which of the following functions belong(s) the set SS ?

A

f(x)=x22e−x+(e−12)e−xf(x)=\frac{x^{2}}{2} e^{-x}+\left(e-\frac{1}{2}\right) e^{-x}

B

f(x)=−x22e−x+(e+12)e−xf(x)=-\frac{x^{2}}{2} e^{-x}+\left(e+\frac{1}{2}\right) e^{-x}

C

f(x)=ex2(x−12)+(e−e24)e−xf(x)=\frac{e^{x}}{2}\left(x-\frac{1}{2}\right)+\left(e-\frac{e^{2}}{4}\right) e^{-x}

D

f(x)=ex2(12−x)+(e+e24)e−xf(x)=\frac{e^{x}}{2}\left(\frac{1}{2}-x\right)+\left(e+\frac{e^{2}}{4}\right) e^{-x}

Answer

f(x)=x22e−x+(e−12)e−xf(x)=\frac{x^{2}}{2} e^{-x}+\left(e-\frac{1}{2}\right) e^{-x}

Explanation

Solution

(A) f(x)=x22e−x+(e−12)e−xf(x)=\frac{x^{2}}{2} e^{-x}+\left(e-\frac{1}{2}\right) e^{-x}
(C)f(x)=ex2(x−12)+(e−e24)e−xf(x)=\frac{e^{x}}{2}\left(x-\frac{1}{2}\right)+\left(e-\frac{e^{2}}{4}\right) e^{-x}